WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2020.19.34 Chantawan Noisri A Simple Numerical Model for Studying Cloud Formation Process in the Tropics : Heated and cooled surface experiments CHANTAWAN NOISRI Department of Mathematics Faculty of Science, Thaksin University Phatthalung, THAILAND Abstract: A simple numerical model for demonstrating local cloud formation processes in the tropics is being developed. The model equations are derived from the fundamental system of partial differential equations of computational fluid dynamics and the deep convection approximation is used to eliminate sound waves. The model domain is two-dimensional with length100 kilometers and height 17.5 kilometers. A non-uniform grid is used with the thinnest layer (100 meters) at the earth's surface and thickest layer (1,300 meters) at the top of the troposphere. The horizontal cell's width one kilometer. The Arakawa-C grid is used for the leapfrog method and forward Euler method. Experiments to study the effects of heating and cooling at the surface and the deep convection approximation in moist air are discussed. The deep convection approximation was found to be unsuitable for a model. The model without the deep convection approximation gives processes expected in the real atmosphere. Key-Words: Cloud formation; Surface heating; Surface cooling; Deep convection; Finite differences. 1. Introduction 2. Model Discription Numerical weather prediction (NWP) models are In the model there is one horizontal dimension, the vertical techniques used to predict the future state of the weather by dimension, and the time dimension. The variables are located solving a set of equations which govern the behavior of the on a staggered grid with stretched grid spacing in the vertical atmosphere. The models used today for research and dimension and constant grid spacing in the horizontal operational weather forecasting are very large and dimension. complicated. They can be understood by professional meteorologists only after long periods of study. The physical The molecular viscosity terms are omitted; the body forces processes in the climate system span an enormous range of are friction at the Earth's surface in the horizontal momentum spatial and temporal scales, and even understanding the equation and gravity in the vertical momentum equation; process of small-scale convective cloud in the atmosphere is heating and cooling of the air occur at the Earth's surface; kinetic energy and potential energy in the temperature equation complicated. The simplified numerical model described in are omitted; the Coriolis force is omitted; no distinction is paper has been done for teaching purposes. made between liquid water and ice; the effects of the moisture A simple numerical model in dry air with horizontal grid on the thermodynamic properties of the air are neglected; rain steps of 1 km has shown by experiments that a time step of is not included; and the deep convection approximation is used. seconds is too large for the model without the deep convection approximation [1]. But with the deep convection The deep convection approximation [4] is approximation one can use a time step of seconds and get reasonable results in numerical experiments on the vertical u w w 0 , (1) movement of air over a heated surface representing a city heat x z 0 z island [2]. Time steps of 0.2 seconds and seconds are 0 used, give stable results for the model without the deep where is calculated from a steady background convection approximation and the model with the deep temperature profile and the assumption of hydrostatic convection approximation, respectively [1]. The same results, a equilibrium in the undisturbed atmosphere. simple numerical model in moist air has shown that a time step of seconds is too large for the model without the deep The steady background temperature profile is an convection approximation but the model with the deep approximation to the annual mean upper air temperatures at convection approximation can use a time step of seconds Bangkok represented by the formula [3]. Tz0 302 0.00675 , (2) where T 0 is in kelvins. E-ISSN: 2224-2678 301 Volume 19, 2020 0.3 0.4 WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2020.19.34 Chantawan Noisri TABLE I. LIST OF SYMBOLS The vertical velocity equation Symbol Description Dw RT T x Horizontal distance Rg (5) Dt z z z Vertical height t Time The wind equation g Acceleration of gravity Du RT T 0.16uu R Gas constant for air R (6) Dt x x 2 ln 0.5z z0 z cv Specific heat of air Air density The last term is horizontal friction, which is applied only in u Horizontal velocity the layer of air of thickness z at the Earth’s surface where the roughness length is z0 w Vertical velocity T Temperature Unsaturated air z 0 Roughness length of the surface The temperature equation qs Surface heating rate per unit area ik, Horizontal and vertical cell indices DT RT u w qs (7) Dt cvv x z c z s Saturated vapor density v Water vapor density The surface heating term is applied only in the layer of air mc Condensed cloud water per unit volume at the Earth’s surface. L Latent heat of condensation of water The water vapor equation Dv uw v (8) 2.1. Governing Equations Dt x z The model equations listed below are derived from the fundamental system of partial differential equations of The condensed cloud water equation computational of fluid dynamics [5]. The temperature and moisture equations used depend on whether or not the air is mc 0 (9) saturated. A simple equation for the saturation vapor density of water as a function of temperature is obtained by integrating the Clausius Clapeyron equation [6] assuming that the latent Saturated air heat of condensation of water vapor is constant and water The temperature equation vapor is an ideal gas. DT W The density equation (10) Dt1 EQ The density equation without the deep convection approximation is The water vapor equation D u w D EW , (3) v (11) Dt x z Dt1 EQ and the density equation with the deep convection The condensed cloud water equation approximation is Dm u w EW c m D w T0 g vc (12) Dt x z1 EQ 0 . (4) Dt T z R E-ISSN: 2224-2678 302 Volume 19, 2020 WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2020.19.34 Chantawan Noisri RT u w qs The initial values of the model variables in each cell are where W , functions of the height of the cell above the Earth’s surface, but c x z c z vv are constant along the horizontal rows of cells. The temperature equation is given by ABT BT Ee / , 3 T 02 k k RTv k 302.211 0.3375 0.16875 , (13) L where k 1, 2,...,25. QL, 2.50 1061Jkg . c v It is assumed the initial values of the velocity, surface heating and amounts of cloud water are zero everywhere. The initial values of the temperature and density satisfy the 2.2. Finite difference set up hydrostatic equation, and the initial values of vapor density are calculated on the assumption that the dew point depression The domain of the model is divided into a 25 100 array of below the initial air temperature is a constant at all heights. cells. The horizontal resolution is one kilometer. A vertical The model variables are fixed at the boundary. coordinate s is used in accordance with the transformation z75 s 25 s2 in order to give thin layers at the Earth’s 2.3. Numerical Experiments surface and thick layers at the top of the troposphere [7]. The model variables are evaluated at points on an Arakawa-C grid. Four different experiments were done on the performance The horizontal velocity is on the left side of the cell, the of the model in various cases as given below to study the vertical velocity is on the bottom of the cell, and other effects of a heated area and cooling area in the middle of the variables are in the center of each cell, as shown in Fig1. domain: o A heated area in the middle of the domain in the model The leapfrog method is used to calculate the model without the deep convection approximation. variables at next time step. The Euler method is used for the o A heated area in the middle of the domain in the model first time step. First and second order finite difference with the deep convection approximation. approximations are used in the modeling of space derivatives. o A cooling area in the middle of the domain in the model In the row of cells at the Earth’s surface one-sided second order without the deep convection approximation. difference approximations to derivatives with respect to z are o A cooling area in the middle of the domain in the model used. with the deep convection approximation. zm26 17,500 s26 25 wik1/2, 1 Yik1/2, 1/2 Yik1/2, 1/2 2 skk 1/ 2 zk1/2 25 k 2 k 5 / 4 m 1/2 uik, 1/2 uik1, 1/2 zm2 100 s2 1 zm1 0 sk1 01 ix ix1 Fig. 1. The model variables are evaluated on an Arakawa-C grid with the stretched grid. The horizontal velocity is on the left side of the cell, the vertical velocity is on the bottom of the cell, and other variables are in the center of each cell. E-ISSN: 2224-2678 303 Volume 19, 2020 WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2020.19.34 Chantawan Noisri 3.